
where \(\lesssim \)  stands for less or equal than, up to universal constants. By the strong law of large numbers and the finite moment Assumption REF  (REF ), we have, as \(n \rightarrow \infty \) . \(\left[ \frac{1}{n} \sum _{i=1}^n |F_q(z_i)|^p \right]^{1/p} \overset{p}{\rightarrow } \left[ \mathbb {E}(|F_q(z_i)|^p) \right]^{1/p}\) . Also for \(p\ge 2\)  and \(m \ge 1\) , \(0 \le m^{1/p-1/2} \le 1\) . Now using Theorem 2.14.2 in [1]}, again under the distribution \(\mathbb {P}^\star \) , we have:
\(&\mathbb {E}^\star \left( \sup _{\theta \in \Theta } \sqrt{m}\Vert G_m^{(b)}(\theta ) - G_n(\theta )\Vert  \right) \lesssim J_{[\,]}(1,\mathcal {F},L_2(\mathbb {P}_n) )\left[ \frac{1}{n} \sum _{i=1}^n |F_q(z_i)|^2 \right]^{1/2},\) 
